DS: Exponents

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DS: Exponents

by beater » Wed Sep 17, 2008 11:24 am
If x and y are positive, is x^3 > y?
(1) sqrt x > y
(2) x > y

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Re: DS: Exponents

by Stuart@KaplanGMAT » Wed Sep 17, 2008 11:48 am
beater wrote:If x and y are positive, is x^3 > y?
(1) sqrt x > y
(2) x > y
Dissecting the question stem: we need info about x and y! When we're dealing with exponents, we always want to remind ourselves that positive fractions can cause wacky things to happen.

(1) sqrtx > y

We could choose x = 100 and y = 2, since 10 > 2.

Is 100^3 > 2? yes!

We could also choose x = 1/4 and y = 1/3, since 1/2 > 1/3.

Is (1/4)^3 > 1/3? no!

We can get a "yes" and a "no": insufficient.

(2) x > y

We could choose x = 100 and y = 2, since 100 > 2.

Is 100^3 > 2? yes!

We could choose x = 1/2 and y = 1/3, since 1/2 > 1/3.

Is (1/2)^3 > 1/3? no!

We can get a "yes" and a "no": insufficient.

Neither statement is sufficient alone, so we need to combine. Accordingly, we now know that:

sqrtx > y

and

x>y

We can still get a "yes" answer by choosing x = 100 and y = 2.

We're no longer allowed to pick x = 1/4 and y = 1/3, since those numbers violate statement (2).

However, we can still pick x = 1/2 and y = 1/3, since sqrt(1/2) > 1/3 and get our "no" answer.

Therefore, we can still get both a "yes" and a "no" answer: even together, not enough information, choose (e).
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Re: DS: Exponents

by lunarpower » Thu Sep 18, 2008 1:33 am
beater wrote:If x and y are positive, is x^3 > y?
(1) sqrt x > y
(2) x > y
stuart's solution for the individual statements is awesome.

here's a theory-based (vs. plugging) method for handling the 2 statements together:

there are 3 possibilities for x: "fraction" (i.e., between 0 and 1), 1 itself, and "normal number" (greater than 1).

if x is a "normal number", then x^3 > x > √x. that will give an automatic "yes" to the question, because x^3 is bigger than both x and √x (each of which is already bigger than y).

if x is 1, then x^3, x, and √x are all 1. therefore, this is another automatic "yes" to the prompt question.

if x is a fraction, though, then √x > x > x^3. therefore, you can get a "no" to the question by selecting a y that's smaller than both √x and x, but BIGGER than x^3.

therefore you can get either a "yes" or a "no", so still insufficient.

ans = (e)
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